Dilithium was named as a extrapolation of another element's name, lithium. Dilithium was of note to Federation science because of qualities as a subspace amplifier, creating transtator current when exposed to subspace fields. It was the only known material which could regulate matter/antimatter reactions, which was why it is used by most species in their warp drives. Different compositions and origins produced slightly different appearances in the crystals, ranging in shape and color. (TOSnovel: Prime Directive)

In 2265, Harry Mudd stated that dilithium crystals were worth 300 times their weight in diamonds. (TOSnovel: Mudd's Angels)

Originally, the lifespan of dilithium was limited until methods were discovered in recrystallizing fractured crystals. This caused a drop in the value of dilithium, which was a precious element around 2275. (TOSnovel: Preserver)

History

After the initial discovery of dilithium, it was discovered that 2-3% of the Quartz on Earth was actually dilithium. This was previously undiscovered because it's difficult to discover the mineral's distinguishing feature -- its extension into the fourth dimension -- through conventional testing. This discovery prompted a gold rush, as many museums dug through their collections and found samples of dilithium. (TOSnovel: Prime Directive)

Pharos, burning after its dilithium deposites were ignited by phaser fire.

In 2254Starfleet engineers constructing Project Pharos on the Pharos siteworld in the Marrat system discovered huge deposits of incredibly pure dilithium; of such purity that had it been mined it would not have required refining. Because Marrat was an open system Starfleet opted to keep the discovery a secret, concerned it could provoke a gold rush, or war! Unfortunately KlingonCommanderKaaj did find out about the discovery and attacked Pharos in response. When the USS Enterprise was sent to investigate why communications had been lost with Pharos Kaaj attacked the starship. To solve the dilemma Captain Christopher Pike had the Enterprise fire on Pharos, igniting the entire planet in a chain reaction, and eliminating the source of conflict. (EVcomic: "The Fires of Pharos")

During a trip back to the 20th century to retrieve two humpback whales, the dilithium crystals of the Klingon ship Kirk and his crew were using began to wear out. Since fission based nuclear reactors was still in use, Captain Spock determined that it might have been possible to obtain high energy photons from such a reactor, and bombard the crystals with them - thus causing dilithium to recrystallize. After obtaining such photons from the United States aircraft carrier Enterprise, Spock and Engineer Montgomery Scott were able to successfully recrystallize dilithium. (TOSmovie: The Voyage Home)

By the 24th century, the process of recrystallizing dilithium had been refined to the point that it had become routine on board Federation vessels. Using a theta-matrix compositor, dilithium could be recrystallized inside the warp reaction chamber. (TNGepisodes: "Family", "Skin of Evil")

In real-life science, element 119 has yet to be discovered. A placeholder name based on its number, "ununennium", has been suggested to document the possible existence of this substance. Some models of theoretical science have stated that heavy elements on this scale may be impossible to synthesize or reproduce, and cannot exist in nature. Subspace science mentioned in Star Trek may account for the contradiction of this being a natural resource on other worlds, as well as any relation to lithium, which is another aspect of this fictional material that has not been borne out in real-life study. The formation of the name has the prefix di- preceding lithium, indicating some sort of twofold increase of the basic lithium. A similar nomenclature is used for trilithium and decalithium.